Vanishing of the second Stiefel–Whitney classes of orientable surfaces

Question

How does one see that the second Stiefel-Whitney class is zero for all orientable surfaces. For $S^2$ this can be seen by $TS^2$ being stably trivial, and for $S^1 \times S^1$ one can use $T (S^1 \times S^1) = TS^1 \times TS^1$, which gives the class in terms of the classes on $TS^1$ (which are all trivial). What about higher genus?

Thanks!

Answer 1

Tangent bundle is stably trivial for any orientable surface (because the normal bundle is trivial).

Answer 2

The second Stiefel-Whitney class of a surface is the mod 2 reduction of the Euler class. Since the Euler characteristic (and hence number) is divisible by 2, $w_2$ is zero.